3.281 \(\int \frac{c+d x^3+e x^6+f x^9}{x (a+b x^3)^3} \, dx\)

Optimal. Leaf size=114 \[ \frac{a^2 b e+a^3 (-f)-a b^2 d+b^3 c}{6 a b^3 \left (a+b x^3\right )^2}+\frac{-a^2 b e+2 a^3 f+b^3 c}{3 a^2 b^3 \left (a+b x^3\right )}-\frac{1}{3} \left (\frac{c}{a^3}-\frac{f}{b^3}\right ) \log \left (a+b x^3\right )+\frac{c \log (x)}{a^3} \]

[Out]

(b^3*c - a*b^2*d + a^2*b*e - a^3*f)/(6*a*b^3*(a + b*x^3)^2) + (b^3*c - a^2*b*e + 2*a^3*f)/(3*a^2*b^3*(a + b*x^
3)) + (c*Log[x])/a^3 - ((c/a^3 - f/b^3)*Log[a + b*x^3])/3

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Rubi [A]  time = 0.15477, antiderivative size = 114, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 30, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.067, Rules used = {1821, 1620} \[ \frac{a^2 b e+a^3 (-f)-a b^2 d+b^3 c}{6 a b^3 \left (a+b x^3\right )^2}+\frac{-a^2 b e+2 a^3 f+b^3 c}{3 a^2 b^3 \left (a+b x^3\right )}-\frac{1}{3} \left (\frac{c}{a^3}-\frac{f}{b^3}\right ) \log \left (a+b x^3\right )+\frac{c \log (x)}{a^3} \]

Antiderivative was successfully verified.

[In]

Int[(c + d*x^3 + e*x^6 + f*x^9)/(x*(a + b*x^3)^3),x]

[Out]

(b^3*c - a*b^2*d + a^2*b*e - a^3*f)/(6*a*b^3*(a + b*x^3)^2) + (b^3*c - a^2*b*e + 2*a^3*f)/(3*a^2*b^3*(a + b*x^
3)) + (c*Log[x])/a^3 - ((c/a^3 - f/b^3)*Log[a + b*x^3])/3

Rule 1821

Int[(Pq_)*(x_)^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_.), x_Symbol] :> Dist[1/n, Subst[Int[x^(Simplify[(m + 1)/n] -
1)*SubstFor[x^n, Pq, x]*(a + b*x)^p, x], x, x^n], x] /; FreeQ[{a, b, m, n, p}, x] && PolyQ[Pq, x^n] && Intege
rQ[Simplify[(m + 1)/n]]

Rule 1620

Int[(Px_)*((a_.) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_.), x_Symbol] :> Int[ExpandIntegrand[Px*(a + b*x)
^m*(c + d*x)^n, x], x] /; FreeQ[{a, b, c, d, m, n}, x] && PolyQ[Px, x] && (IntegersQ[m, n] || IGtQ[m, -2]) &&
GtQ[Expon[Px, x], 2]

Rubi steps

\begin{align*} \int \frac{c+d x^3+e x^6+f x^9}{x \left (a+b x^3\right )^3} \, dx &=\frac{1}{3} \operatorname{Subst}\left (\int \frac{c+d x+e x^2+f x^3}{x (a+b x)^3} \, dx,x,x^3\right )\\ &=\frac{1}{3} \operatorname{Subst}\left (\int \left (\frac{c}{a^3 x}+\frac{-b^3 c+a b^2 d-a^2 b e+a^3 f}{a b^2 (a+b x)^3}+\frac{-b^3 c+a^2 b e-2 a^3 f}{a^2 b^2 (a+b x)^2}+\frac{-b^3 c+a^3 f}{a^3 b^2 (a+b x)}\right ) \, dx,x,x^3\right )\\ &=\frac{b^3 c-a b^2 d+a^2 b e-a^3 f}{6 a b^3 \left (a+b x^3\right )^2}+\frac{b^3 c-a^2 b e+2 a^3 f}{3 a^2 b^3 \left (a+b x^3\right )}+\frac{c \log (x)}{a^3}-\frac{1}{3} \left (\frac{c}{a^3}-\frac{f}{b^3}\right ) \log \left (a+b x^3\right )\\ \end{align*}

Mathematica [A]  time = 0.105827, size = 104, normalized size = 0.91 \[ \frac{\frac{\frac{a \left (-a^2 b^2 \left (d+2 e x^3\right )-a^3 b \left (e-4 f x^3\right )+3 a^4 f+3 a b^3 c+2 b^4 c x^3\right )}{\left (a+b x^3\right )^2}+2 \left (a^3 f-b^3 c\right ) \log \left (a+b x^3\right )}{b^3}+6 c \log (x)}{6 a^3} \]

Antiderivative was successfully verified.

[In]

Integrate[(c + d*x^3 + e*x^6 + f*x^9)/(x*(a + b*x^3)^3),x]

[Out]

(6*c*Log[x] + ((a*(3*a*b^3*c + 3*a^4*f + 2*b^4*c*x^3 - a^2*b^2*(d + 2*e*x^3) - a^3*b*(e - 4*f*x^3)))/(a + b*x^
3)^2 + 2*(-(b^3*c) + a^3*f)*Log[a + b*x^3])/b^3)/(6*a^3)

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Maple [A]  time = 0.015, size = 147, normalized size = 1.3 \begin{align*} -{\frac{{a}^{2}f}{6\,{b}^{3} \left ( b{x}^{3}+a \right ) ^{2}}}+{\frac{ae}{6\,{b}^{2} \left ( b{x}^{3}+a \right ) ^{2}}}-{\frac{d}{6\,b \left ( b{x}^{3}+a \right ) ^{2}}}+{\frac{c}{6\,a \left ( b{x}^{3}+a \right ) ^{2}}}+{\frac{\ln \left ( b{x}^{3}+a \right ) f}{3\,{b}^{3}}}-{\frac{c\ln \left ( b{x}^{3}+a \right ) }{3\,{a}^{3}}}+{\frac{2\,af}{3\,{b}^{3} \left ( b{x}^{3}+a \right ) }}-{\frac{e}{3\,{b}^{2} \left ( b{x}^{3}+a \right ) }}+{\frac{c}{3\,{a}^{2} \left ( b{x}^{3}+a \right ) }}+{\frac{c\ln \left ( x \right ) }{{a}^{3}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((f*x^9+e*x^6+d*x^3+c)/x/(b*x^3+a)^3,x)

[Out]

-1/6*a^2/b^3/(b*x^3+a)^2*f+1/6*a/b^2/(b*x^3+a)^2*e-1/6/b/(b*x^3+a)^2*d+1/6/a/(b*x^3+a)^2*c+1/3/b^3*ln(b*x^3+a)
*f-1/3*c*ln(b*x^3+a)/a^3+2/3*a/b^3/(b*x^3+a)*f-1/3/b^2/(b*x^3+a)*e+1/3/a^2/(b*x^3+a)*c+c*ln(x)/a^3

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Maxima [A]  time = 0.953155, size = 174, normalized size = 1.53 \begin{align*} \frac{3 \, a b^{3} c - a^{2} b^{2} d - a^{3} b e + 3 \, a^{4} f + 2 \,{\left (b^{4} c - a^{2} b^{2} e + 2 \, a^{3} b f\right )} x^{3}}{6 \,{\left (a^{2} b^{5} x^{6} + 2 \, a^{3} b^{4} x^{3} + a^{4} b^{3}\right )}} + \frac{c \log \left (x^{3}\right )}{3 \, a^{3}} - \frac{{\left (b^{3} c - a^{3} f\right )} \log \left (b x^{3} + a\right )}{3 \, a^{3} b^{3}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((f*x^9+e*x^6+d*x^3+c)/x/(b*x^3+a)^3,x, algorithm="maxima")

[Out]

1/6*(3*a*b^3*c - a^2*b^2*d - a^3*b*e + 3*a^4*f + 2*(b^4*c - a^2*b^2*e + 2*a^3*b*f)*x^3)/(a^2*b^5*x^6 + 2*a^3*b
^4*x^3 + a^4*b^3) + 1/3*c*log(x^3)/a^3 - 1/3*(b^3*c - a^3*f)*log(b*x^3 + a)/(a^3*b^3)

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Fricas [A]  time = 1.36771, size = 374, normalized size = 3.28 \begin{align*} \frac{3 \, a^{2} b^{3} c - a^{3} b^{2} d - a^{4} b e + 3 \, a^{5} f + 2 \,{\left (a b^{4} c - a^{3} b^{2} e + 2 \, a^{4} b f\right )} x^{3} - 2 \,{\left ({\left (b^{5} c - a^{3} b^{2} f\right )} x^{6} + a^{2} b^{3} c - a^{5} f + 2 \,{\left (a b^{4} c - a^{4} b f\right )} x^{3}\right )} \log \left (b x^{3} + a\right ) + 6 \,{\left (b^{5} c x^{6} + 2 \, a b^{4} c x^{3} + a^{2} b^{3} c\right )} \log \left (x\right )}{6 \,{\left (a^{3} b^{5} x^{6} + 2 \, a^{4} b^{4} x^{3} + a^{5} b^{3}\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((f*x^9+e*x^6+d*x^3+c)/x/(b*x^3+a)^3,x, algorithm="fricas")

[Out]

1/6*(3*a^2*b^3*c - a^3*b^2*d - a^4*b*e + 3*a^5*f + 2*(a*b^4*c - a^3*b^2*e + 2*a^4*b*f)*x^3 - 2*((b^5*c - a^3*b
^2*f)*x^6 + a^2*b^3*c - a^5*f + 2*(a*b^4*c - a^4*b*f)*x^3)*log(b*x^3 + a) + 6*(b^5*c*x^6 + 2*a*b^4*c*x^3 + a^2
*b^3*c)*log(x))/(a^3*b^5*x^6 + 2*a^4*b^4*x^3 + a^5*b^3)

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Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((f*x**9+e*x**6+d*x**3+c)/x/(b*x**3+a)**3,x)

[Out]

Timed out

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Giac [A]  time = 1.07024, size = 173, normalized size = 1.52 \begin{align*} \frac{c \log \left ({\left | x \right |}\right )}{a^{3}} - \frac{{\left (b^{3} c - a^{3} f\right )} \log \left ({\left | b x^{3} + a \right |}\right )}{3 \, a^{3} b^{3}} + \frac{3 \, b^{4} c x^{6} - 3 \, a^{3} b f x^{6} + 8 \, a b^{3} c x^{3} - 2 \, a^{4} f x^{3} - 2 \, a^{3} b x^{3} e + 6 \, a^{2} b^{2} c - a^{3} b d - a^{4} e}{6 \,{\left (b x^{3} + a\right )}^{2} a^{3} b^{2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((f*x^9+e*x^6+d*x^3+c)/x/(b*x^3+a)^3,x, algorithm="giac")

[Out]

c*log(abs(x))/a^3 - 1/3*(b^3*c - a^3*f)*log(abs(b*x^3 + a))/(a^3*b^3) + 1/6*(3*b^4*c*x^6 - 3*a^3*b*f*x^6 + 8*a
*b^3*c*x^3 - 2*a^4*f*x^3 - 2*a^3*b*x^3*e + 6*a^2*b^2*c - a^3*b*d - a^4*e)/((b*x^3 + a)^2*a^3*b^2)